Unit 12 - Grade 11-12 Physics

Oscillations, Waves, and Sound

Study simple harmonic motion, mass-spring systems, simple pendulums, restoring forces, wave properties, superposition, resonance, standing waves, sound intensity, harmonics, beats, and the Doppler effect.

Lesson roadmap

What Students Should Master in This Unit

Oscillations explain repeating motion such as springs, pendulums, vibrating strings, and sound sources. This unit helps students connect simple harmonic motion, wave diagrams, equations, sound behavior, interference patterns, resonance, and musical harmonics.

Analyze SHM

Use restoring force, amplitude, period, frequency, phase, angular frequency, spring systems, and pendulums.

Describe wave motion

Use amplitude, wavelength, frequency, period, phase, wave speed, and medium correctly.

Analyze wave interactions

Predict reflection, superposition, constructive interference, destructive interference, and standing waves.

Apply sound physics

Connect pitch, loudness, intensity, decibels, resonance, harmonics, beats, and Doppler shifts.

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Jump to a Topic

Simple Harmonic Motion Wave Basics Wave Types Wave Speed Wave Equation Wave Behaviors Superposition and Interference Standing Waves Sound Basics Intensity and Decibels Strings and Air Columns Beats and Doppler Effect Simulation Labs Lab Skills Worked Examples Practice Problems
Repeating motion around equilibrium

1. Simple Harmonic Motion and Oscillations

Simple harmonic motion (SHM) happens when an object experiences a restoring force that points back toward equilibrium and is proportional to displacement. Springs, small-angle pendulums, vibrating strings, and many sound sources depend on this idea.

Simple harmonic motion in springs and pendulums SHM: restoring force pulls the system back toward equilibrium equilibrium spring force = -kx x mass-spring oscillator θ restoring component simple pendulum, small angles
Springs use Hooke's law. Pendulums behave like SHM only for small angles, where gravity creates an approximately proportional restoring effect.
Restoring force F = -kx Negative means the force points opposite displacement.
SHM condition a = -ω2x Acceleration points toward equilibrium.
Position model x(t) = A cos(ωt + φ) Sine can also be used depending on the starting point.
Angular frequency ω = 2πf = 2π/T Measured in rad/s.
Mass-spring period T = 2π√(m/k) More mass increases period; stiffer spring decreases period.
Simple pendulum period T = 2π√(L/g) Works best for small angles, usually below about 15°.
Maximum speed vmax = Aω Occurs as the object passes equilibrium.
Maximum acceleration amax = Aω2 Occurs at maximum displacement.
Spring energy E = 1/2 kA2 Total mechanical energy in an ideal spring oscillator.

Important SHM Topics

  • Equilibrium position, amplitude, period, frequency, phase, and angular frequency.
  • Restoring force and why the negative sign matters.
  • Energy transfer between kinetic energy and elastic or gravitational potential energy.
  • Damped oscillations, driven oscillations, and resonance.
  • How SHM connects to wave motion: a wave can be viewed as many particles oscillating while energy travels.
Core idea: SHM is the bridge into waves. Before students study traveling waves, they should understand that each point in a medium can oscillate around equilibrium.
Energy moving through space

2. Wave Basics

A wave is a disturbance that transfers energy. In many waves, particles in the medium vibrate around equilibrium while the wave pattern travels forward.

Basic wave measurements A wave transfers energy while the medium vibrates A λ wave direction equilibrium
Amplitude is the maximum displacement from equilibrium, and wavelength is the distance for one complete cycle.
Amplitude A = maximum displacement Larger amplitude usually means more energy.
Wavelength λ = distance for one cycle Crest to crest, compression to compression, or node pattern repeat.
Frequency f = cycles / time Measured in hertz, Hz.
Period T = time for one cycle Measured in seconds.
Frequency-period relation f = 1/T Also T = 1/f.
Phase same phase = same point in cycle Important for interference.
Core idea: Waves carry energy. The medium particles usually oscillate around their starting positions instead of traveling with the wave all the way across.
How particles move compared with wave direction

3. Types of Waves

Waves can be classified by how the particles move and whether a material medium is required.

Transverse and longitudinal waves Classify waves by particle motion transverse: particles move perpendicular longitudinal: particles move parallel
Transverse waves vibrate perpendicular to travel direction; longitudinal waves vibrate parallel to travel direction.
Wave Type Particle Motion Examples
Transverse waveParticles vibrate perpendicular to wave directionWave on a rope, light waves, water surface waves partly
Longitudinal waveParticles vibrate parallel to wave directionSound in air, compressions in a spring
Mechanical waveRequires a material mediumSound, water waves, waves on strings
Electromagnetic waveDoes not require a mediumLight, radio waves, microwaves, X-rays
PulseSingle disturbanceOne flick on a rope
Periodic waveRepeating disturbanceContinuous vibration from a speaker or oscillator

Transverse Wave Parts

  • Crest: highest point above equilibrium.
  • Trough: lowest point below equilibrium.
  • Amplitude: maximum displacement from equilibrium.
  • Wavelength: distance between matching points, such as crest to crest.

Longitudinal Wave Parts

  • Compression: region where particles are close together.
  • Rarefaction: region where particles are spread apart.
  • Wavelength: distance from compression to compression or rarefaction to rarefaction.
How fast a disturbance travels

4. Wave Speed

Wave speed depends on the medium. Changing frequency does not usually change wave speed in the same medium; instead wavelength changes.

Wave speed equation In one medium, frequency and wavelength trade off wavelength λ v = fλ travel speed
In the same medium, wave speed is set by medium properties, so increasing frequency usually shortens wavelength.
Main wave speed equation v = fλ Speed equals frequency times wavelength.
Speed from distance and time v = d/t Works for wave pulses and travel-time problems.
Speed on a string v = √(FT/μ) FT is tension, μ is mass per length.
Linear density μ = m/L Mass per unit length of a string.
Sound speed in air v ≈ 331 + 0.60TC TC is air temperature in °C.
Room-temperature sound v ≈ 343 m/s Typical value near 20°C.
Mathematical model of a sinusoidal wave

5. Wave Equation and Phase

A sinusoidal wave can be described by a function that depends on position and time. Grade 11-12 students do not always need the full equation, but it helps explain phase and interference.

Wave phase Phase tells where a wave is within its cycle phase shift y(x,t) = A sin(kx - ωt)
Two waves with the same frequency can still interfere differently if their phases are shifted.
Traveling wave y(x,t) = A sin(kx - ωt) Wave moving in the positive x direction.
Wave number k = 2π/λ Spatial cycles in radians per meter.
Angular frequency ω = 2πf Temporal cycles in radians per second.

Reading Wave Graphs

  • On a displacement-position graph, wavelength is measured horizontally.
  • On a displacement-time graph, period is measured horizontally.
  • Amplitude is measured vertically from equilibrium to crest or trough.
How waves interact with boundaries

6. Wave Behaviors

When waves meet boundaries, openings, or new media, they can reflect, refract, diffract, transmit, or absorb.

Common wave behaviors Waves change behavior at boundaries and openings reflection refraction diffraction
Reflection, refraction, and diffraction explain many wave behaviors, including echoes, bending, and spreading through openings.
Behavior Meaning Example
ReflectionWave bounces off a boundaryEcho from a wall.
RefractionWave changes speed and direction in a new mediumSound bending in temperature layers.
DiffractionWave spreads around edges or through openingsSound heard around a doorway.
TransmissionWave passes into or through a mediumSound through a wall.
AbsorptionWave energy converts into internal energyAcoustic foam reducing echoes.
Diffraction rule: Diffraction is strongest when the opening or obstacle size is comparable to the wavelength.
When waves overlap

7. Superposition and Interference

When waves overlap, their displacements add. After passing through each other, the waves continue moving as before.

Superposition and interference Overlapping waves add their displacements constructive: larger result destructive: smaller result
Constructive interference occurs when waves reinforce; destructive interference occurs when opposite displacements reduce the result.
Superposition principle ytotal = y1 + y2 Add displacements at the same point.
Constructive interference waves add Crest with crest or compression with compression.
Destructive interference waves cancel partly or fully Crest with trough or compression with rarefaction.
Path difference for constructive ΔL = mλ m = 0, 1, 2, ... for sources in phase.
Path difference for destructive ΔL = (m + 1/2)λ For sources in phase.
Phase difference one wavelength = 360 degrees Half wavelength means 180 degrees out of phase.
Stable vibration patterns

8. Standing Waves and Resonance

A standing wave forms when two waves of the same frequency and amplitude travel in opposite directions and interfere. Resonance occurs when a system is driven at one of its natural frequencies.

Standing waves and resonance Standing waves have fixed nodes and moving antinodes nodes antinode resonance occurs when driving frequency matches a natural frequency
Standing waves are stable patterns with nodes that stay still and antinodes that vibrate with maximum amplitude.
Nodes zero displacement points Always still in an ideal standing wave.
Antinodes maximum displacement points Largest oscillation amplitude.
String fixed at both ends L = nλ/2 n = 1, 2, 3, ...
String harmonic frequencies fn = nv/(2L) All integer harmonics allowed.
Fundamental frequency f1 = v/(2L) Lowest natural frequency for a string fixed at both ends.
Harmonic relation fn = nf1 For strings and open-open pipes.
Common mistake: A node is not the same thing as a crest. A node is a point that does not move in a standing wave.
Longitudinal pressure waves

9. Sound Basics

Sound is a mechanical longitudinal wave. In air, it travels as moving compressions and rarefactions created by vibrating objects.

Sound as a longitudinal pressure wave Sound travels through compressions and rarefactions compression rarefaction higher f = higher pitch
Sound needs a medium because pressure variations move through particles that compress and spread apart.
Sound speed in air near 20°C v ≈ 343 m/s Sound speed changes with temperature and medium.
Pitch higher f = higher pitch Frequency controls perceived pitch.
Loudness larger amplitude = louder sound Related to intensity and pressure variation.

Human Hearing

  • Typical human hearing range is about 20 Hz to 20,000 Hz.
  • Infrasound is below 20 Hz.
  • Ultrasound is above 20,000 Hz.
  • Sound cannot travel through a vacuum because it needs a medium.
Energy per area per time

10. Sound Intensity and Decibels

Sound intensity measures power carried by sound through a unit area. The decibel scale is logarithmic, which means small decibel changes can represent large intensity changes.

Sound intensity and decibels Sound spreads out, so intensity decreases with distance I = P/(4πr2) β = 10 log(I/I0) +10 dB means intensity x10 doubling distance gives 1/4 intensity
The decibel scale is logarithmic, so intensity ratios matter more than simple addition.
Intensity I = P/A Unit: W/m2.
Spherical spreading I = P/(4πr2) For sound spreading equally in all directions.
Decibel level β = 10 log(I/I0) I0 = 1.0 × 10-12 W/m2.
10 dB increase intensity x 10 A 20 dB increase means intensity x 100.
Distance doubles intensity becomes 1/4 For spherical spreading in open space.
Threshold of hearing I0 = 1.0 × 10-12 W/m2 Reference intensity for decibels.
Musical instruments and harmonics

11. Strings and Air Columns

Musical instruments work by creating standing waves. Strings, open pipes, and closed pipes have different boundary conditions, so their allowed wavelengths and frequencies differ.

Strings and air column harmonics Boundary conditions decide the allowed harmonics string: L = λ/2 open-open pipe closed-open pipe: only odd harmonics
Strings and open-open pipes support all integer harmonics, while closed-open pipes support only odd harmonics.
System Allowed Wavelengths Allowed Frequencies
String fixed at both endsλn = 2L/nfn = nv/(2L), n = 1, 2, 3, ...
Open-open pipeλn = 2L/nfn = nv/(2L), n = 1, 2, 3, ...
Closed-open pipeλn = 4L/nfn = nv/(4L), n = 1, 3, 5, ...

Boundary Conditions

  • Closed end of an air column: displacement node, pressure antinode.
  • Open end of an air column: displacement antinode, pressure node.
  • Fixed end of a string: displacement node.
  • Higher harmonics have higher frequency and shorter wavelength.
Changing sound patterns

12. Beats and the Doppler Effect

Beats happen when two close frequencies interfere and create alternating loud and quiet sound. The Doppler effect happens when source and observer motion changes the observed frequency.

Beats and Doppler effect Frequency patterns can change through interference or motion beats: loud-soft envelope Doppler: compressed wavefronts ahead
Beats come from interference between close frequencies; the Doppler effect comes from relative motion between source and observer.
Beat frequency fbeat = |f1 - f2| Number of loud-soft cycles per second.
Doppler effect idea approaching -> higher observed f Receding source or observer gives lower observed frequency.
Stationary observer, moving source f' = f[v/(v -/+ vs)] Use minus when source moves toward observer, plus when away.
Moving observer, stationary source f' = f[(v ± vo)/v] Use plus when observer moves toward source, minus when away.
Quick check: If source and observer move closer together, the observed frequency should be higher. If they move farther apart, it should be lower.
Simulation labs

13. Simulation Labs for This Unit

These official PhET simulations help students visualize oscillation period, spring stiffness, pendulum length, wave speed, amplitude, frequency, interference, sound, harmonics, and Fourier wave building.

Wave simulation lab workflow Use simulations to connect controls, wave shape, and explanation Change m, k, L, f, A, tension Observe λ, speed, nodes, pattern Explain use wave equations A good simulation answer links a changed variable to the observed wave pattern.
Wave simulations are most useful when students control one variable and explain the resulting pattern with frequency, wavelength, and speed.
Masses and Springs

Explore Hooke's law, spring constant, mass, amplitude, equilibrium position, and how period changes in a vertical spring oscillator.

Lab idea: change mass or spring stiffness and compare the observed period with T = 2π√(m/k).
Open Simulation
Pendulum Lab

Study how pendulum length, gravity, release angle, damping, and mass affect periodic motion.

Lab idea: test which variables affect period and compare results with T = 2π√(L/g).
Open Simulation
Wave on a String

Explore amplitude, frequency, tension, damping, wave speed, reflection, and standing wave behavior on a string.

Lab idea: increase tension and observe how wave speed and standing-wave patterns change.
Open Simulation
Sound

Visualize compressions, rarefactions, air pressure variation, frequency, amplitude, and how speakers create sound waves.

Lab idea: compare how frequency changes pitch and wavelength when sound speed is constant.
Open Simulation
Wave Interference

Investigate constructive interference, destructive interference, diffraction, double-source patterns, and wave fronts.

Lab idea: change source spacing and wavelength, then describe how the interference pattern changes.
Open Simulation
Fourier: Making Waves

Build complex wave shapes using harmonics and see how multiple sine waves combine by superposition.

Lab idea: add harmonics one at a time and observe how the resulting wave shape changes.
Open Simulation
Investigation skills

14. Oscillations, Waves, and Sound Lab Skills

Oscillation and wave labs usually require careful measurement of time, length, frequency, amplitude, mass, spring constant, and boundary conditions. Students should also describe patterns clearly using diagrams and labels.

Waves and sound lab measurements Measure length, time, frequency, and boundary conditions λ travel distance d v = d/t or v = fλ record frequency, period, wavelength, string length, air temperature, and uncertainty
Professional wave lab work clearly labels the measured wavelength, frequency, timing method, and boundary conditions.

Common Labs

  • Mass-spring period lab comparing measured period with T = 2π√(m/k).
  • Simple pendulum lab testing how length, amplitude, and mass affect period.
  • Hooke's law lab measuring spring constant from force and stretch.
  • Wave speed on a string lab using frequency and wavelength.
  • Standing waves on a string or spring investigation.
  • Sound speed measurement using echo timing or resonance tubes.
  • Open-pipe and closed-pipe resonance lab.
  • Beat frequency lab using tuning forks or tone generators.
  • Interference and diffraction simulation lab.
  • Fourier synthesis lab for complex wave shapes.

Useful Measurements

  • Mass in kilograms and spring constant in N/m.
  • Pendulum length in meters and release angle in degrees.
  • Frequency in hertz.
  • Period in seconds.
  • Wavelength in meters.
  • String length or air-column length in meters.
  • Travel time for echoes or pulses.
  • Amplitude or relative loudness.
  • Temperature of air for sound speed corrections.
Lab warning: For resonance tubes, remember end correction can shift the effective length slightly beyond the open end.
Worked examples

15. Worked Examples

SHM Example: Mass-spring period

A 0.50 kg mass is attached to a spring with k = 200 N/m. Find the period.

T = 2π√(m/k) = 2π√(0.50/200) = 0.314 s.

SHM Example: Simple pendulum period

A pendulum has length 0.80 m. Use g = 9.8 m/s2. Find its small-angle period.

T = 2π√(L/g) = 2π√(0.80/9.8) = 1.79 s.

Example 1: Frequency from period

A wave has period 0.020 s. Find frequency.

f = 1/T = 1/0.020 = 50 Hz.

Example 2: Wave speed

A wave has frequency 12 Hz and wavelength 0.80 m. Find speed.

v = fλ = (12)(0.80) = 9.6 m/s.

Example 3: Wavelength of sound

A 440 Hz sound travels in air at 343 m/s. Find wavelength.

λ = v/f = 343/440 = 0.780 m.

Example 4: Speed on a string

A string has tension 80 N and linear density 0.020 kg/m. Find wave speed.

v = √(FT/μ) = √(80/0.020) = 63.2 m/s.

Example 5: Fundamental on a string

A string of length 0.75 m has wave speed 120 m/s. Find fundamental frequency.

f1 = v/(2L) = 120/[2(0.75)] = 80 Hz.

Example 6: Closed pipe fundamental

A closed-open pipe has length 0.50 m. Use sound speed 343 m/s. Find fundamental frequency.

For a closed pipe, f1 = v/(4L) = 343/[4(0.50)] = 171.5 Hz.

Example 7: Beat frequency

Two tuning forks produce 256 Hz and 260 Hz. Find beat frequency.

fbeat = |260 - 256| = 4 Hz.

Example 8: Sound intensity level

A sound has intensity 1.0 × 10-6 W/m2. Find decibel level.

β = 10 log(I/I0) = 10 log[(1.0 × 10-6)/(1.0 × 10-12)] = 60 dB.

Example 9: Echo distance

An echo returns after 0.80 s. Use v = 343 m/s. How far away is the wall?

The sound travels to the wall and back, so d = vt/2 = (343)(0.80)/2 = 137 m.

Example 10: Doppler check

A siren moves toward a stationary observer. Is the observed frequency higher or lower?

Higher. Wave fronts are compressed in front of the moving source.

Independent practice

16. Practice Problems

Try each problem first. Then open the answer check and compare formulas, units, and wave reasoning.

1. A wave has period 0.25 s. Find frequency.

Answer

f = 1/T = 1/0.25 = 4.0 Hz.

2. A wave has frequency 5.0 Hz and wavelength 2.0 m. Find speed.

Answer

v = fλ = (5.0)(2.0) = 10 m/s.

3. A sound wave travels at 343 m/s with frequency 686 Hz. Find wavelength.

Answer

λ = v/f = 343/686 = 0.500 m.

4. A wave travels 24 m in 3.0 s. Find speed.

Answer

v = d/t = 24/3.0 = 8.0 m/s.

5. What type of wave is sound in air?

Answer

A mechanical longitudinal wave.

6. What happens to wavelength if frequency increases while speed stays constant?

Answer

Wavelength decreases because v = fλ.

7. A string has tension 100 N and μ = 0.010 kg/m. Find wave speed.

Answer

v = √(FT/μ) = √(100/0.010) = 100 m/s.

8. A 1.2 m string is fixed at both ends and wave speed is 96 m/s. Find fundamental frequency.

Answer

f1 = v/(2L) = 96/[2(1.2)] = 40 Hz.

9. For the string in problem 8, find the third harmonic frequency.

Answer

f3 = 3f1 = 3(40) = 120 Hz.

10. An open-open pipe has length 0.85 m. Find fundamental frequency using v = 343 m/s.

Answer

f1 = v/(2L) = 343/[2(0.85)] = 202 Hz.

11. A closed-open pipe has length 0.85 m. Find fundamental frequency using v = 343 m/s.

Answer

f1 = v/(4L) = 343/[4(0.85)] = 101 Hz.

12. Which harmonics exist in a closed-open pipe?

Answer

Only odd harmonics: n = 1, 3, 5, ...

13. Two waves arrive in phase. Is interference constructive or destructive?

Answer

Constructive interference.

14. Two in-phase sources have path difference 2λ. Is the interference constructive or destructive?

Answer

Constructive, because ΔL = mλ.

15. Two in-phase sources have path difference 1.5λ. Is the interference constructive or destructive?

Answer

Destructive, because ΔL = (m + 1/2)λ.

16. Two tones have frequencies 440 Hz and 445 Hz. Find beat frequency.

Answer

fbeat = |445 - 440| = 5 Hz.

17. Sound intensity increases by a factor of 100. How many decibels does the level increase?

Answer

Increase = 10 log(100) = 20 dB.

18. A source emits power equally in all directions. If distance doubles, what happens to intensity?

Answer

Intensity becomes one-fourth as large because I is proportional to 1/r2.

19. A wall produces an echo 1.2 s after a clap. Use v = 343 m/s. Find wall distance.

Answer

d = vt/2 = (343)(1.2)/2 = 206 m.

20. A sound source moves away from a listener. What happens to observed frequency?

Answer

Observed frequency decreases because the wave fronts are stretched out.

21. A 0.25 kg mass oscillates on a spring with k = 100 N/m. Find the period.

Answer

T = 2π√(m/k) = 2π√(0.25/100) = 0.314 s.

22. A simple pendulum has length 1.20 m. Estimate its period using g = 9.8 m/s2.

Answer

T = 2π√(L/g) = 2π√(1.20/9.8) = 2.20 s.

23. In SHM, where is speed greatest: at equilibrium or at maximum displacement?

Answer

Speed is greatest at equilibrium because energy is mostly kinetic there.

Final review

17. What to Know Before Moving On

  • Simple harmonic motion occurs when restoring force is proportional to displacement and points toward equilibrium.
  • For a spring oscillator, F = -kx and T = 2π√(m/k).
  • For a small-angle pendulum, T = 2π√(L/g).
  • In ideal SHM, maximum speed occurs at equilibrium and maximum acceleration occurs at maximum displacement.
  • Damping removes energy; resonance occurs when driving frequency matches a natural frequency.
  • Waves transfer energy without transporting matter all the way with the wave.
  • Wave speed is v = fλ.
  • Frequency and period are related by f = 1/T.
  • Transverse waves vibrate perpendicular to travel direction; longitudinal waves vibrate parallel.
  • Sound is a mechanical longitudinal wave and needs a medium.
  • Wave speed depends mainly on the medium.
  • Superposition means overlapping wave displacements add.
  • Constructive interference makes larger amplitude; destructive interference reduces amplitude.
  • Standing waves contain nodes and antinodes.
  • Strings and open-open pipes allow all integer harmonics.
  • Closed-open pipes allow only odd harmonics.
  • Beat frequency is the difference between two close frequencies.
  • The Doppler effect changes observed frequency when source and observer move relative to each other.
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